C₂×#20251871 #27079778

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#20251871

Splittings

Sylow

C₄₇ ⋊ C₂×#2889

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₂²×2₋¹⁺⁴×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂²×2₋¹⁺⁴	D₄₇
hypocenter	C₄₇	C₂×#2889
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×2₋¹⁺⁴

4770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 3008

A minimal faithful permutation representation — 5 generators on 81 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(53,64)(57,70)(61,75)(65,77)(68,74)(71,80)(76,78)(79,81);(50,53)(51,57)(52,61)(54,65)(55,64)(56,68)(58,70)(59,71)(60,74)(62,76)(63,75)(66,78)(67,77)(69,79)(72,80)(73,81);(50,52,55,63)(51,56,58,60)(53,61,64,75)(54,66,67,62)(57,68,70,74)(59,73,72,69)(65,78,77,76)(71,81,80,79);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(50,51,54,59,55,58,67,72)(52,60,62,73,63,56,66,69)(53,57,65,71,64,70,77,80)(61,74,76,81,75,68,78,79);(48,49)

Conjugacy classes & character table

1646 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1033068);

C2×#20251871 #27079778